Problem 28 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. $$\begin{array}{r}9 \frac{4}{9} \\\\+1 \frac{1}{6} \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The sum is \(10 \frac{11}{18}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions for ease of calculation. \(9 \frac{4}{9}\) becomes \(\frac{9 \times 9 + 4}{9} = \frac{81 + 4}{9} = \frac{85}{9}\). Similarly, \(1 \frac{1}{6}\) becomes \(\frac{1 \times 6 + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6}\).

Step 2: Find a Common Denominator

To add the fractions, they must have a common denominator. The denominators are 9 and 6; their least common multiple is 18. Convert \(\frac{85}{9}\) to \(\frac{85 \times 2}{18} = \frac{170}{18}\), and \(\frac{7}{6}\) to \(\frac{7 \times 3}{18} = \frac{21}{18}\).

Step 3: Add the Fractions

Now, add the fractions with the common denominator: \(\frac{170}{18} + \frac{21}{18} = \frac{191}{18}\).

Step 4: Convert the Result Back to a Mixed Number

Divide the numerator by the denominator to convert back to a mixed number. \(191 \div 18 = 10\) with a remainder of 11. Thus, the result is \(10 \frac{11}{18}\).

Key Concepts

Improper Fractions

When dealing with mixed numbers, sometimes it's easier to handle calculations by converting them into improper fractions first. An improper fraction is when the numerator—this is the number on top—is larger than the denominator, which is the number on the bottom. For example, when you have a mixed number like \(9 \frac{4}{9}\), you want to convert it by multiplying the whole number by the denominator and then adding the numerator to the result. This gives you \(\frac{85}{9}\). Doing this makes it much easier to add, subtract, multiply, or divide fractions later.

Common Denominator

In order to perform addition or subtraction with fractions, you must first ensure that all fractions have the same denominator, called a common denominator. This means the denominators are equal, allowing us to combine the numerators directly. For instance, \(\frac{85}{9}\) and \(\frac{7}{6}\) can't be directly added. But if we find a common denominator, like 18, we can convert \(\frac{85}{9}\) to \(\frac{170}{18}\) and \(\frac{7}{6}\) to \(\frac{21}{18}\). Once the denominators are the same, you can easily add the fractions by adding their numerators, resulting in \(\frac{191}{18}\). Takeaway: same denominator equals easy arithmetic!

Least Common Multiple

Finding a common denominator often involves finding the least common multiple (LCM) of the denominators you are working with. The LCM is the smallest number that can be divided evenly by both denominators without leaving a remainder. To find the LCM of 9 and 6, you list the multiples of each number. The first number that appears in both lists is your LCM, which is 18 in this case. By converting both fractions to have this new denominator, you simplify the process of adding or subtracting fractions.

• List the multiples of each denominator.
• Find the smallest multiple common to both lists.
• Use this common multiple as the new denominator.

Once you have the LCM, the process of adding becomes simpler!

Converting Mixed Numbers

Converting improper fractions back to mixed numbers is the final step in many calculations involving fractions. Once you've completed the addition or subtraction of your fractions, you may end up with an improper fraction like \(\frac{191}{18}\). To convert this fraction back into a mixed number, you divide the numerator by the denominator. In this example, divide 191 by 18, which goes evenly 10 times, leaving a remainder of 11. This gives you the mixed number \(10 \frac{11}{18}\). This final result is much easier to interpret and understand compared with an improper fraction.
Steps to convert include:

• Divide the numerator by the denominator.
• The quotient becomes the whole number.
• The remainder forms the new numerator over the original denominator.

This process makes it simple to transform your fraction back into a friendly format!